### The Academic Failure to Understand Rebalancing

Perhaps the most universally accepted investing principle is to periodically rebalance one’s portfolio. Advisors have been drilled that rebalancing results in some combination of improved performance and reduced risk. Unfortunately, this precept is the byproduct of imperfect mathematics; the benefits of rebalancing are far smaller than what advisors have come to believe.

The best technical article on portfolio rebalancing – by a very wide margin – was not published in a finance journal. It was written by A. J. Wise and published in the British Actuarial Journal in 1996. Meanwhile, dozens of articles on rebalancing in finance journals are filled with mathematics that is riddled with obvious flaws. The conclusions most of these articles draw from their mathematics are not only trivial but also so vaguely stated that it is difficult to discern exactly what the conclusions are and what their relationship is to the practical issue of portfolio rebalancing.

Why does academic finance fill its articles with mathematical sound and fury that signifies so little? What is wrong with the field of “mathematical finance” that it can produce such inferior work, published in peer-reviewed journals, with so little relevance for anyone who wishes to know whether portfolio rebalancing provides a benefit?

I will explore these questions after discussing the results of the academic finance papers and comparing them with Wise’s results.

**The simple results of the finance papers**

It is challenging to examine the academic finance papers on rebalancing^{[1]} in a non-technical publication, given the arcane nature and impenetrability of those papers’ mathematics and language. Therefore, I will paraphrase them in a simple way. Rest assured that this paraphrased version is completely analogous to the analyses in the papers themselves.

Let’s say someone offers you a bet on the toss of a fair coin (50-50 chance of heads or tails). If it comes up heads, your bet will be doubled (100% return); if tails, your bet will be halved (–50% return). In other words, a dollar bet on the coin toss will become two dollars if it comes up heads, but only 50 cents if it comes up tails.

Suppose you bet a dollar on the first coin toss and let it ride. If the coin is tossed some number of times – say, n times – and it comes up heads m of those times, a little easy math shows that your initial dollar will become two raised to the m power times one half raised to the (n – m) power dollars (that is, $2^{m}0.5^{n-m}). If there were 10 tosses in total and m was six (six heads), your initial dollar will become $2^{6}0.5^{4}, or four dollars.

But if the 10 tosses produced only five heads, you’ll end up with the same one dollar you started with, for a zero rate of return.

In the long run, the law of large numbers says that the proportion of heads will gradually become one half. So in the long run, m will come closer and closer to half of n. That means that the longer you play the game, the more likely it is that you’ll just wind up with the same dollar you started with and realize only a zero rate of return.

**Kelly’s trick**

However, you can “trick” this game, using a strategy devised by John Kelly in 1956. The first time you play you bet only half your dollar and keep the other 50 cents in reserve. Win or lose, you play the same strategy each time – you “rebalance” your portfolio. Each time you play, you place 50% of your portfolio on the bet and keep the other 50% in cash. For example, if you win the first toss you’ll have $0.50+0.50x2 = $1.50. So on the second toss you’ll put 75 cents in reserve and bet the other 75 cents.

It can now be shown using easy math that after n tosses, m of which are heads, your portfolio will have a number of dollars equal to 3^{n}/2^{2n-m}. In the long run, m will get closer and closer to one half of n. So this number will converge to $3^{n}/2^{1.5n}, which is much greater than one dollar.

For example, if you play 20 times and m, the number of heads, is the expected 10, then you’ll have $3^{20}/2^{1.5x20}, which is $3.25 – much more than the $1 you would have had if you had bet 100% of your portfolio every time. It’s also greater than what you would have had if you had initially bet only half of your dollar and then let it ride – a 50/50 buy-and-hold strategy. That strategy would have yielded, after 20 tosses, half of which were heads, an end result of $1 also.

It would seem that the strategy of rebalancing on every toss to a 50-50 cash/bet mix outperforms both the 100% bet strategy and the 50-50 buy-and-hold strategy.

As we shall see, however, this is woefully incomplete; yet it is pretty much all that the finance articles say. Virtually all of the most-cited finance papers on the subject bury this simple result under layers of stochastic differential calculus, accompanied by suggestive but inadequately defined terminology such as “diversification return,” “volatility pumping” and “excess return.” Then they claim or imply, based on this result alone, that rebalancing is the superior strategy.